Restricting directions for Kakeya sets

TL;DR

This paper establishes the equivalence between the Kakeya maximal conjecture and the $ ext{o}$-Kakeya maximal conjecture, and provides improved bounds on the Hausdorff dimension of $ ext{o}$-Kakeya sets, advancing understanding in geometric measure theory.

Contribution

It proves the equivalence of two major conjectures in Kakeya theory and improves bounds on the Hausdorff dimension of $ ext{o}$-Kakeya sets.

Findings

Proved the equivalence of Kakeya and $ ext{o}$-Kakeya maximal conjectures.

Established a new lower bound for the Hausdorff dimension of $ ext{o}$-Kakeya sets.

Extended previous results by Keleti and Mathé on Kakeya conjectures.

Abstract

We prove that the Kakeya maximal conjecture is equivalent to the $\Omega$-Kakeya maximal conjecture. This completes a recent result in [2] where Keleti and Math{\'e} proved that the Kakeya conjecture is equivalent to the $\Omega$-Kakeya conjecture. Moreover, we improve concrete bound on the Hausdorff dimension of a $\Omega$-Kakeya set : for any Bore set $\Omega$ in S n--1 , we prove that if X $\subset$ R n contains for any e $\in$ $\Omega$ a unit segment oriented along e then we have dX $\ge$ 6 11 d$\Omega$ + 1 where dE denotes the Hausdorff dimension of a set E.